Hall-closure and products of Fitting classes

Abstract

The Fitting class (of finite, soluble, groups), Of, is said to be Hall ir-closed (where IT is a set of primes) if whenever G is a group in g and H is a Hall w-subgroup of G, then H belongs to gf. In this paper, we study the Hall w-closure of products of Fitting classes. Our main result is a characterisation of the Hall w-closed Fitting classes of the form 2f#©^ (where <& m denotes the so-called smallest normal Fitting class), subject to a restriction connecting ir with the characteristic of g. We also characterise those Fitting classes g (respectively, ©) such that 3E#g (respectively, 5E)#3E) is Hall ir-closed for all Fitting classes X. In each case, part of the proof uses a concrete group construction. As a bonus, one of these constructions also yields a "cancellation result" for certain products of Fitting classes.